Comparison of algorithms for solving a bi-level toll setting problem

Vyacheslav Kalashnikov, Fernando Camacho, Ronald Askin, Nataliya Kalashnykova

Research output: Contribution to journalArticlepeer-review

21 Scopus citations


In this paper, we consider the problem of assigning optimal tolls to the arcs of a multi-commodity transportation network. The problem is formulated as a bi-level mathematical program where the upper level is managed by a firm that raises revenues from tolls set on arcs of the network and the lower level is represented by a group of car users traveling along the cheapest paths with respect to a generalized travel cost. The problem can be interpreted as finding an equilibrium among tolls generating high revenues and tolls attracting customers. We describe the bi-level programming model and discuss the underlying assumptions. Next, we propose and evaluate four algorithms based on different principles to solve the toll optimization problem. In order to solve this problem efficiently, we first reformulate it as a standard mathematical program and describe a penalty-function algorithm for its solution. The algorithm is well-founded and its convergence is established. We then detail a proposed quasi-Newton-type algorithm, a gradient approximation-based algorithm, and a direct method making use of the Nelder-Mead flexible simplex search. The results of the numerical experiments support the algorithms' robustness. ICIC International

Original languageEnglish (US)
Pages (from-to)3529-3549
Number of pages21
JournalInternational Journal of Innovative Computing, Information and Control
Issue number8
StatePublished - Aug 2010


  • Bi-level programming
  • Gradient approximation method
  • Nelder-Mead algorithm
  • Optimum toll setting problem
  • Penalty function methods
  • Quasi-Newton-type method

ASJC Scopus subject areas

  • Software
  • Theoretical Computer Science
  • Information Systems
  • Computational Theory and Mathematics


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